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    Geometry of fully coordinated, two-dimensional percolation

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    We study the geometry of the critical clusters in fully coordinated percolation on the square lattice. By Monte Carlo simulations (static exponents) and normal mode analysis (dynamic exponents), we find that this problem is in the same universality class with ordinary percolation statically but not so dynamically. We show that there are large differences in the number and distribution of the interior sites between the two problems which may account for the different dynamic nature.Comment: ReVTeX, 5 pages, 6 figure

    Brownian Occupation Measures, Compactness and Large Deviations

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    In proving large deviation estimates, the lower bound for open sets and upper bound for compact sets are essentially local estimates. On the other hand, the upper bound for closed sets is global and compactness of space or an exponential tightness estimate is needed to establish it. In dealing with the occupation measure L_t(A)=\frac{1}{t}\int_0^t{\1}_A(W_s) \d s of the dd dimensional Brownian motion, which is not positive recurrent, there is no possibility of exponential tightness. The space of probability distributions M1(Rd)\mathcal {M}_1(\R^d) can be compactified by replacing the usual topology of weak convergence by the vague toplogy, where the space is treated as the dual of continuous functions with compact support. This is essentially the one point compactification of Rd\R^d by adding a point at \infty that results in the compactification of M1(Rd)\mathcal M_1(\R^d) by allowing some mass to escape to the point at \infty. If one were to use only test functions that are continuous and vanish at \infty then the compactification results in the space of sub-probability distributions M1(Rd)\mathcal {M}_{\le 1}(\R^d) by ignoring the mass at \infty. The main drawback of this compactification is that it ignores the underlying translation invariance. More explicitly, we may be interested in the space of equivalence classes of orbits M~1=M~1(Rd)\widetilde{\mathcal M}_1=\widetilde{\mathcal M}_1(\R^d) under the action of the translation group Rd\R^d on M1(Rd)\mathcal M_1(\R^d). There are problems for which it is natural to compactify this space of orbits. We will provide such a compactification, prove a large deviation principle there and give an application to a relevant problem.Comment: Minor revision. To appear in the "Annals of Probability